# Let vector u , v , and w be vectors such that vector u+v+w=0 if |u|=3, |v|=4 and |w| =5 then u.v+v.w+w.u=?

Solution:    Since   $\vec{u}+\vec{v}+\vec{w}=0$ , we have

$\left | \vec{u}+\vec{v}+\vec{w} \right |^{2}=0\\ \\ \Rightarrow \hspace{1cm}\left | \vec{u} \right |^{2}+\left | \vec{v} \right |^{2}+\left | \vec{w} \right |^{2}+2(\vec{u}\cdot \vec{v}+\vec{v}\cdot \vec{w}+\vec{w}\cdot \vec{u})=0\\ \\ \Rightarrow \hspace{1cm}9+16+25+2(2(\vec{u}\cdot \vec{v}+\vec{v}\cdot \vec{w}+\vec{w}\cdot \vec{u}))=0\\ \\ \Rightarrow \hspace{1cm}(\vec{u}\cdot \vec{v}+\vec{v}\cdot \vec{w}+\vec{w}\cdot \vec{u})=-25$

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